Market Risk

FRM Part II - Book 1

A coherent risk measure is a risk metric that satisfies a set of properties, such as subadditivity, positive homogeneity, and translation invariance.

What is Subadditivity?

• Subadditivity: The risk measure of a portfolio is less than or equal to the sum of the risk measures of its components.

A coherent risk measure is a risk metric that satisfies a set of properties, such as subadditivity, positive homogeneity, and translation invariance.

What is Positive homogeneity ?

• Positive homogeneity: Scaling the portfolio by a positive factor scales the risk measure by the same factor.

A coherent risk measure is a risk metric that satisfies a set of properties, such as subadditivity, positive homogeneity, and translation invariance.

What is translation invariance?

• Translation invariance: Adding a constant to the portfolio does not change the risk measure.

A generally coherent risk measure tends to involve increasingly sophisticated weighting functions. Which of the following is a suitable replacement for the equal weights in the 'average VaR' to estimate any risk measure?

Weights appropriate to risk measure being estimated.

The 'average VaR' is a simple method for estimating the VaR of a portfolio or assets by averaging the VaR of each component, with equal weights assigned to each component. However, because different components may have varying levels of risk, correlation, or impact on the overall portfolio, this approach may not be appropriate for all types of risk measures. Therefore, the appropriate weights must be chosen based on the underlying component characteristics and the specific risk measure being evaluated to estimate any risk measure using a weighted approach

Coherent risk measures are fundamental in the field of risk management, providing a consistent framework for assessing the risk of financial portfolios. The key to estimating coherent risk measures lies in the:

Ability to estimate quantiles.

Many coherent risk measures, particularly Expected Shortfall (ES), depend on accurately estimating the tail of the loss distribution. Expected Shortfall is defined as the average loss beyond a specified quantile (VaR). Therefore, before ES can be calculated, the corresponding quantile (VaR) must first be estimated.

An analyst has gathered the following information about a portfolio that has normally distributed geometric returns:

Mean: 10%
Standard Deviation: 40%
Portfolio: 100 million

What is the 95% lognormal VAR for this portfolio?

$42.8 million.

Assume you are dealing with a stock “ A” that displays a highly negatively skewed distribution comprised of the past 260-days returns. Suppose you have P1 = A and P2 = -A, meaning P1 is long stock A and P2 is short stock A. Which statement is most likely to be accurate about a 99% VaR?

Correct Answer is #1 - |VaR (P1)| > |VaR (P2)|.

Given that the return distribution of stock A is negatively skewed, it displays a long left tail. This implies large potential losses for a long position and large potential gains for a short position. Therefore, |VaR (P1)| will be expected to be higher.

A data scientist is analyzing a dataset and wants to determine the distribution of his data. The scientist decides to use a QQ plot in his analysis. Which of the following statements about QQ plots is correct?

Option 3 - QQ plots are useful in determining if a dataset follows a normal distribution.

A QQ plot is a graphical tool used in statistics to help visualize how a dataset is distributed. It plots the quantiles of the dataset against the quantiles of a standard normal distribution. If the dataset follows a normal distribution, the points on the QQ plot will approximately lie on a straight line. This is because the quantiles of the dataset will match closely with the quantiles of the standard normal distribution. Therefore, a QQ plot is a powerful tool for visually checking the assumption of normality in a dataset.

A small prop trading firm has only 80 days of historical returns. They want to estimate VaR at 97.5% but find the traditional historical simulation percentile approach too coarse. The head quant suggests a non-parametric density estimation method.

What key benefit does this non-parametric approach offer the firm?

Option 2 - Flexible percentile selection.

The key benefit of using a non-parametric density estimation method like Kernel Density Estimation (KDE) is that it creates a smooth, continuous distribution from the limited historical data. This allows for flexible percentile selection.

With only 80 data points, using a direct percentile method (like taking the 2nd smallest return for a 97.5% VaR) is very sensitive to individual data points and doesn't give a very precise estimate. KDE interpolates between the data points, creating a much smoother distribution from which you can read off any percentile you want with greater precision than the raw data would allow.

Which statement correctly captures the essence of a correlation-weighted approach within the historical simulation framework?

Option 2 - Correlation-weighted historical simulation aims to address a key limitation of standard historical simulation: its reliance on historical correlations, which may not be representative of current or future market conditions. This approach adjusts the historical returns in a way that preserves the individual volatilities of the assets but alters their co-movements (correlations) to reflect a desired or estimated correlation structure. This is often done using techniques like Cholesky decomposition.

Even though bootstrapping has numerous advantages, the bootstrap estimates are associated with a little bias or error. Which of the following presents an error of bootstrapping?

Re-sampling variability

In the context of bootstrapping, re-sampling variability refers to the error that arises from the fact that we take only a finite number of bootstrap re-samples (denoted by B) from our original sample, rather than an infinite number. This limitation is inherent to the bootstrapping process, as it is practically impossible to take an infinite number of re-samples. As a result, our bootstrap estimates are subject to a certain degree of variability that is directly related to the number of re-samples taken. The larger the number of re-samples, the lower the re-sampling variability, and vice versa. However, even with a large number of re-samples, the re-sampling variability can never be completely eliminated, thus contributing to the overall error in the bootstrap estimates.

Paul is using the age-weighted historical simulation approach to estimate the VaR of a stock portfolio, with a historical sample size of 100 days and a decay factor of 0.96. Over the recent past, the portfolio has registered the following returns.

Determine the weight on the return earned 45 days ago

0.006751.

An analyst performing a historical simulation to measure the VaR of a portfolio uses a volatility-weighted approach. One month ago, the actual return of the asset was 5% and its daily volatility estimate was 2%. If the current daily volatility is 1.5%, calculate the volatility-adjusted return.

0.0375

Understanding the correlation between portfolio assets is crucial in risk management, especially when calculating Value at Risk (VaR). The degree of correlation significantly impacts the overall risk profile of the portfolio. If portfolio assets are perfectly correlated, portfolio VaR will equal:

Undiversified VaR

When the assets in a portfolio are perfectly correlated, the portfolio's VaR would be equal to the undiversified VaR. This is because perfect correlation implies that all assets in the portfolio move in the same direction at the same time. Therefore, there is no benefit from diversification, and the portfolio's risk is equivalent to the sum of the individual asset risks, which is the definition of undiversified VaR.

A large investment bank's risk management team is implementing a VaR mapping framework for their diversified fixed-income portfolio containing corporate bonds, government securities, and mortgage-backed securities across multiple currencies and credit ratings. The portfolio exhibits significant exposure to both systematic market movements and idiosyncratic issuer-specific events. The risk manager must decide on the appropriate mapping methodology that will most effectively capture the dual nature of risks while maintaining computational efficiency for daily risk reporting. The team recognizes that the chosen approach will fundamentally determine how well their VaR estimates reflect the portfolio's true risk profile during both normal and stressed market conditions.

Which approach would most appropriately balance the capture of general and specific risks in this mapping process?

Map each security to principal component factors plus issuer-specific idiosyncratic risk components

The principal component approach with issuer-specific idiosyncratic components most effectively captures both general and specific risks in VaR mapping. Principal components capture the systematic risk factors that drive correlated movements across the fixed-income universe (general risk), while the issuer-specific idiosyncratic components capture the unique risks associated with individual securities (specific risk). This approach provides the optimal balance between comprehensiveness and computational efficiency, as principal components reduce dimensionality while preserving most of the variance in the data, and the idiosyncratic components ensure that security-specific events are not overlooked in the risk calculation.

Adding more general risk factors to a VaR model will most likely:

2 - Decrease the size of specific risks.

The addition of more general risk factors to a Value at Risk (VaR) model will most likely decrease the size of specific risks. Specific risk, also known as idiosyncratic risk or residual risk, refers to the risk associated with individual assets in a portfolio, which is not correlated with market risk. By incorporating more general risk factors into the model, the model becomes more comprehensive and can better capture the overall risk dynamics of the portfolio. This, in turn, reduces the portion of risk that is unexplained by the model, i.e., the specific risk. For instance, if a model only considers the duration of the portfolio, it will have a larger specific risk compared to a model that considers both duration and credit risk. Therefore, adding more general risk factors to the model helps define the risk more accurately and reduces the specific or residual risk.

A forward rate agreement is a type of forward contracts that allows the contracting parties to make transactions in a locked interest rate at some future date. The buyer in this contract locks in the borrowing rate and the seller locks in the lending rate at some future date.

Which statement is true about such a contract if the spot rate is higher than the forward rate at the time of the transaction?

Option 3 - The buyer will benefit by borrowing at a lower locked-in rate, while the seller will be worse off by lending at a lower rate.

In a Forward Rate Agreement (FRA), the buyer (or the 'long') locks in a borrowing rate, while the seller (or the 'short') locks in a lending rate. If the spot rate at the time of the transaction is higher than the forward rate, the buyer will benefit. This is because the buyer will make payments at a lower rate than the prevailing market rate. On the other hand, the seller will be worse off because they will be lending at a lower rate than the current market rate. This scenario is advantageous for the buyer because they have effectively hedged against the risk of rising interest rates. Conversely, the seller, who may have anticipated a fall in interest rates, will be at a disadvantage as they are locked into lending at a lower rate.